Landmark Representation and Dynamic Mediation

Abstract

Models of human cognition and learning have been largely static,
using simple causal relations to capture the dynamic character. Dynamic properties
are difficult to represent on paper, which is fundamentally a static medium.
In this interactive paper, I will describe a interactive framework of representation
and mediation expressed in a multi-agent computer simulation system that
has
the potential
to capture more effectively these dynamics. The representational and process
framework will be introduced in the domains of color, orientation, number,
and spatial representation of simple shapes. The basic concepts of representation
and mediation
presented here may have applicability in broader contexts.

Introduction

There has been a continuing debate about the representation of
things that vary on a continuum with discrete concepts. Many of the
aspects of the world vary in ways that are perceived as continuous. However
these continua are perceived by organisms with neural systems consisting
of discrete neurons. So the challenge is to build a simulation system that
is able to represent points on a continuum and changes in
values along a continuum with discrete elements.

Rosch (1973) conducted a set of studies of color perception
across a variety of cultures, that supported the concept of color "prototypes".
So even though color hue is physically a continuously changing quantity (at
the level of analysis of human perception), the perception of color involves
color "prototypes", discrete entities that mark out the space of
color hue. This seems to be pretty strong evidence for the discrete representation
of color; however people do in fact perceive a continuum of color hue. How
can we reconcile these two aspects of color perception?

In this interactive paper, I will introduce a framework
for representation, called "landmark representation" and a framework
for processing,
called "interactive mediation", that characterize a distributed activity
approach to thinking and learning. These new frameworks will be illustrated
with NetLogo Web files,
which are HTML5 files that can be accessed over the web that allow the reader
to take an active role in exploring the dynamic nature of these frameworks.
The files were created in NetLogo (Wilensky,
1999), a powerful multi-agent modeling environment that is cross-platform,
freely available and allows simulations to be saved as HTML5 web files that can
be accessed and run over the Web. In each case, if you are reading this paper through
the web, you can click on a link or figure to start up one of these files.
Thes files are fairly large, so may take a while to load. If you are reading this paper in print
format, you may want to access the web version at the same time (or at some
point, if it is not possible right now) and try out the web files. The URL for
the web version is: http://pages.ucsd.edu/~jalevin/landmark-mediation/

Discrete landmarks and continuous representation.

But there are obvious landmarks, suggested in
fact by the discrete way that the eye perceives color, with three distinct
color
receptors
in the
retina.
This discreteness is also represented by the naming system we have for colors,
as shown in Figure 0.2.

Figure 0.2: Discrete color names

So the challenge is to represent a continuum like color in a
way that captures both the continuous nature of the continuum and the discrete
nature of the categorical system associated with it, in an integrated way.

Figure 0.3 Discrete and continuous color

For example, how can we represent the color "red"?
We can have a concept for "red", and the activity level value of
that concept represents how intense the red is. The activity level is a number
that varies from 0 (complete inactivity) to some maximum number (maximum activity
level). Now if we have two color landmarks, let's say for "red"
and for "green", then we can represent not just those two landmark
colors, but any intermediate color by different ratios of activity level values
for the two
landmarks.
So a color half-way between red and green would be represented by the situation
in which the activity level of the two color landmarks are equal, as illustrated
by the
web file shown in Figure 1 below. In that figure, the two landmarks, Red and
Green,
are represented as two circles, and
the activity level of each is shown by the number within the circle for each
landmark and by the intensity of the color of each circle. Click
on
the
figure
below
or this
link to
start
the
web file so that you can actively see how different ratios of activity level can
represent the colors that
are between red and green.

Figure 1: An web file that shows the representation of a color
continuum with two color landmarks.

So, a color half way between red and green is yellow. As you
move the marker closer to the Red landmark, the Red landmark becomes more active
relative to the Green landmark, and the color hue represented is more orange.
As you move the marker closer to the "Green" landmark, the "Green" landmark
becomes relatively more active,
and yellowish-green hues are represented. So with these two landmarks, we can
represent a continuum of color hues

Figure 2: An web file that shows the representation of a color
plane with three color hue landmarks.

As you move the "X" around in the plane defined by
the three color landmarks, the color of the "X" changes to indicate
the color hue represented by the particular ratio of activity levels of the
landmarks. As in Figure 1, the activity level of each of the three landmarks
is indicated by the number inside each circle and by the intensity of the color
of each circle.

This landmark representation framework is also not limited to
three landmarks. A difference between those with more expert knowledge of color
and
those will
less expertise could well be the larger number of color hue landmarks a more
expert person has, and we can imagine a developmental process in
which
a relative novice with few color landmarks acquires color hue expertise through
the acquisition of more and more color hue landmarks. So a relative novice
would
have color hue landmarks for common colors such as yellow and orange, while
an expert would have acquired more sophisticated color landmarks such as teal
(a dark to medium greenish blue) and mauve (a pale bluish purple). Both would
be able to represent the continuum of color hue, but the expert would be able
to make finer discriminations between colors.

A landmark representation of orientation.

It isn't just the color continuum that can be represented with
landmarks. There is also a continuum of directional orientations, which in
a plane are captured by the
principle compass points of north, east, south, and west. This directional
orientation continuum can be represented by these four landmarks, with intermediate
orientations
represented by the simultaneous activity of two or more of these orientation
landmarks. Figure
3 is an web file that shows this orientation representational system, representing
the orientation of northeast.

Figure 3: An web file that shows the representation of planar
orientation, with four orientation landmarks.

A landmark representation of the number line.

Moving to a more abstract domain, a prototypical continuum is
the number line. We can capture the infinite number of numbers represented
by the number line by a finite number of numeric landmarks. Figure 4 is
an
web file that shows this number line representational system, with the number
500 represented as equal activity levels of the landmarks 0 and 1000.

Figure 4: An web file that shows the representation of the number 500 with
equal activity levels of landmarks "0" and "1000".

Recent research on children's representation of number concepts
by Siegler and Opfer (2003) has shown that young
children represent numbers on the positive number line
in an exponential
way, then move gradually to a linear way as they develop. Siegler and Opfer
have argued for two different representational systems, with development shifting
from
one to the other with increasing mathematical experience (Siegler & Opfer,
2003). With the"landmark" representational system presented here,
we can model this same developmental process with just one representational
system, by having young children starting with just a few
numeric
landmarks
and gradually acquiring more and more numeric landmarks. In particular, if
they acquire the landmarks featured by the place value system early in the
process (1 10 100 1000 10000 ...), then their numeric performance would look
like they had an exponential model. As they acquired more and more landmarks,
especially integer landmarks, their performance would look more and more linear.

This numeric landmark web file allows you to choose other sets
of landmarks, including "1 2 3" and "1 10 100". Figure
5 illustrates of the landmark set "1 2 3" with activity levels of
the three landmarks representing 2.5.

Figure 5: A representation of the number 2.5 as the differential activity of
three numeric landmarks, 1, 2, & 3.

Now an alternative representation of that same number 2.5 would
be to have equal activity levels for the two closest landmarks, 2 and 3. In
fact, with this representational system, there are an infinite number of different
representations
for the same number. Figure 6 shows an
web file that allows you to select a
number to be represented, and then to see the different ways that that number
is represented by changing the standard deviation of a Gaussian distribution
with the represented number as the mean value.

Figure 6: A representation of the number 2.5 with a different set of activity
levels
of three landmarks: 1, 2, & 3.

So this landmark representational system not only allows the
representation of any value on a continuum, but it also allows the representation
of the variability
or uncertainty of the value. It can represent more certain knowledge about
a value by activity of immediately neighboring landmarks; it can represent
less certain knowledge about a value by the activity of more than just the
neighboring landmarks.

If you compare Figure 7 with Figure 2, you can see that a smaller
variability leads to a darker hue. If you click on Figure
7, you can see the
effect of changing the standard deviation for any given color hue.

Generalization and differentiation.

When presented with two or more stimuli, organisms are presented
with two different challenges. One is to identify the similarities between
the different stimuli; another simultaneous challenge is to identify differences.
In one case, the challenge is to generalize across multiple instances of a
stimulus; in the case, the challenge is to differentiate between different
stimuli. With the landmark representational framework presented here, there
is a simple
basic mechanism for modeling these generalization and differentiation processes.
With two (or more) instances of a value on a continuum represented by a set
of landmarks, a model for generalization is to simply combine the activations
of the landmarks for the different values to be generalized across in an additive
way (through combined excitation). In an analogous way, a model for differentiation
is to combine
the activations of the landmarks
for the different values to be differentiated by subtracting the activity levels
of all the landmarks representing one of the values from the activity levels
of the
all the landmarks of the other value (by a combination of excitation and inhibition).
Figure
8 shows this for numbers represented by numeric landmarks.

In this numeric domain, the generalization model basically determines
the mean value of the two numbers being generalized. The differentiation model
is a kind of "rounding", pushing the first value toward the nearest landmark
away from the second value.

We have seen how values on a number line can be represented by
landmarks and their differential activity. Let's now look again at lines (actually
line segments) as spatial entities. There are a number of mathematical and
geometric definitions of a line (and line segment). What would be a landmark
representation of a line segment? Figure 10 shows one such representation.

Figure 10: A landmark representation of a line segment.

In this landmark definition, a line is defined as having three
landmark subparts, a mid and two ends. But what are the other things in Figure
10? There are three new entities, all of which are spatial mediations among
the subparts and between the line concept and subparts. "Near" is
a strong proximity mediator, that operates to bring together its immediate
neighbors
into a common place. "Together" is a weaker proximity mediator, that
operates to bring together its immediate neighbors, but not as strongly as
the "near"
mediator. The third mediator "apart" is a mediator that operates
to move its immediate neighbors apart from itself (and thus from each other).
What would
we predict if all of these mediators operated simultaneously on the line and
its landmark subparts? Figure
11 shows an web file that implements these mediators.

Figure 11: An web file that implements a landmark subpart definition of a line
segment with spatial mediators.

It is difficult to convey in a static medium the dynamic properties
of this interactive mediational representation of a line segment. There are
two properties that are clear from interaction with the dynamic representation
of the web file: self-repair and self-organization. While interacting with
the web file, if you select some subpart or some mediator and move it then let
it go, you will see the line segment "self-repair". That is, the
moved component will operate on its immediate neighbors to restore its specified
relationships,
and the other components will simultaneously operate to restore their specified
relationships. The line segment will restore itself, although in a different
position and/or orientation. The second, related, property of this dynamic
representation is "self -organization". When interacting with the
web file, if you select the "Scramble" button, all of the components
will be moved to a random location within the space. Figure 12 shows a scrambled
state of the
line concept.

Figure 12: A randomized state of the line segment representation.

Then, selecting the "Go" button leads to a dynamic
interaction of all the components, leading to the stable state shown in Figure
13.

Figure 13: A self-organized line segment representation.

Another interesting property of this interactive mediator representation
of a line segment is the definition of length of the line, which can be manipulated
by the line length slider. The length of a line segment in this representation
is a "dynamic stability" property, the balance between the strength of attraction
exerted by the "together" mediators and the strength of repulsion exerted by
the "apart" mediator. The stronger the "apart" mediator, the longer the line
segment's length; the stronger the "together" mediators, the shorter the length

The letter "A" is defined as having three lines as
components, with the spatial mediator "near" specified between pairs
of landmark subparts of these three lines. The end of line 1 is "near" the
end of line 2, thus specifying the apex of the letter "A". The cross-line
is line 3, with one end "near" the
the mid of line 1 and the other end near the mid of line 2. Figure 15 shows
the stable state reached when all the mediators simultaneously operate.

Figure 15: The stable state of the interactive mediation of the
letter "A".

The "straightness" of the lines, represented explicitly
in figures 10 through 13, is "built-in" to the concept of "line" in
figures 14 and 15, as a kind of proceduralization of the mediation. The
letter "A" is both self-repairing and self-organizing, in the same
way described above for a simple line. If one of the components of the "A" is
moved, the component is moved back into place by its interactions with the
other components. At the same time, the other components are moved in a lesser
degree, leading to translation and/or rotation of the overall letter in the
process of self-repair. If the components of the letter "A" are randomized
with the "Scramble" button, the components interact with each other
to "self-organize" the spatial definition of the letter, which will
appear in a random orientation and location. Figure 16 shows some screenshots
of the
process
of
self-organizing.

Figure 16: Steps in the process of the self-organization of a scrambled letter
"A".

This spatial representation and the ones that follow are structural
descriptions of spatial shapes (Palmer, 1999). In fact, this representation
of an "A" is very similar to Figure 8.2.15 in Palmer's book "Vision
Science" (Palmer,
1999, p. 394), which is reproduced with permission as Figure 16.1.

Figure 16.1: Palmer (1998) structural description of the letter A.

It is also similar
to a set of representations that I sketched out
in an unpublished paper many years ago (Levin, 1973). The representation
of the letter A is shown in Figure 16.2.

Figure 16.2: Levin (1973) structural representation of the
letter A.

The major difference between the representations in this paper
and previous structural representations
is that
this is an interactive structural representation with multiple simultaneously
operating mediators acting to maintain or organize the spatial shape.

Oriented shapes.

While it is useful to have an orientation-free representation
of a letter, to assist in the recognition and production of the letter in non-canonical
orientations, it is also useful to have an orientation-specified representation.
Figure
17 shows a representation of an upright-oriented letter A, created by
adding the orientation mediator "above" between the apex of the A
and the mid of the cross-point.

Figure 17: An interactive mediational representation of an upright A.

If the letter A is not upright, the "above" mediator
will operate to rotate the letter, through its impact on its immediate neighbors,
the intersection
of the apex of the A (represented by the mediator "near1") and the
mid point landmark of the cross-bar line 3. The "above" mediator pushes the
"near" mediator upward and the "mid" mediator downward, and through those pushes,
exerts torque that rotates the letter into an upright orientation.

"Invisible" lines as structural constraint
mediators.

Let us look at the representation of other letters. Suppose we
wanted to represent the letter "L". Figure
18 shows a simple representation of the letter.This representation is
the same as the one that I specified many years ago in the appendix of
the Levin (1973) paper. Once the web file starts, select "an-overly-simple-L" from
the selection box at the bottom left, then click "Go".

Figure 18: An overly simple representation of the letter L.

However, if you move one of the components of this overly simple
representation, you will find that it does not self-repair
into the letter L, since the proximity mediator "near" keeps the two ends of
the two lines together, but not at a right angle. Instead it operates as a
"hinge", allowing the two lines to take any orientation in relation to each
other. This representation captures part of the spatial character of the letter
"L" but not all of it.

One thing that can be learned from the representation of the
"A" shown above is the power of triangles to provide stability. So
one way to represent a stable right angle that is characteristic of the letter "L"
is to add an "invisible" line that creates a triangle. This invisible
line is sort of like the "construction" lines used in geometric proofs. Figure
19 shows an web file that captures the property of the right angle in an "L".
Once the web file start, select "L" from the "Which_shape" set of choices at
the bottom left.

Figure 19: A self-organizing, self-repairing representation of the letter L.

Bi-stable representations.

Sometimes a randomized letter "L" will self-organize
into a normal letter "L" as shown in Figure 19, perhaps in a different
location or orientation,
and sometimes it will self-organize into a
reversed letter "L" as
shown in Figure 20.

Figure 20: A representation of a reversed "L".

If you move components of this reversed "L", in many case it
will "self-repair" back into a reversed "L". You can make quite large changes
without leading to a reorganization into a normal "L". However, if you move
one of the ends of one of the lines to the other side of the other end, then
this representation will reorganize into a normal "L".

Reversal isn't the only bi-stable state. We can define the letter
"X" as two straight lines for which the middles are near each other,
and the appropriate ends are apart from each other. This is shown in Figure
21. Once the web file starts, select "X" from the "Which_shape" choices at
the bottom left.

Figure 21: A representation of the letter "X".

There are two different stable states for such a representation.
One is the one in which the two line segments cross each other; another is
one in which the two line segments approach each other and then diverge from
each other, forming two "V" shapes that are joined. These two stable
states are shown in Figure 22.

Figure 22: Two stable states of the representation of the letter "X".

More complex lines and approximations to curves.

So far, we've only represented letters that involve straight
lines. Do we need to introduce the concept of continuous curves? One way to
represent continuous curves is by adding more and more landmarks to a line
segment, landmarks that allow the line segment to "flex" at those
landmark points. Figure
22.1 shows a more complex line, with five subpart landmarks
instead of the three subparts landmarks of the simple line segment shown above.

Figure 22.1: A more complex line segment.

When you click the "Scramble" button, you can see the line flex
while it is self-organizing, sometime needing to "unfold" in order to create
linearity.

Now we can use this more complex line to construct letter shapes
that require a curved line, where this complex line serves as an approximation
to the curved line, with the landmark points serving to "mark" the continuous
curve.

Figure
23 shows a representation of the letter "D" that involves the
simple representation of a line for the vertical line component, and a more
complex
line segment to represent the curved component. Once the web file starts, select "D" from the "Which_shape" selection
box at the bottom left.

Figure 23: A representation of the letter "D" with a more complex
line segment approximating a curve.

We can use the same complex line to create a representation of
the letter "B". One additional function for the invisible line is
to keep the two loops of the "B" on the same side of the simple line,
as shown in Figure
24. After the web file starts, select "B" from the "Which_shape" choices
at the bottom left.

Figure 24: A representation of the letter "B", with an invisible
line keeping the loops on the same side.

This issue of "same side" vs. "opposite side" becomes clearer
when we represent the letter "S" and the digit "3". Figure 25 shows a representation
for the letter "S" and FIgure 26 shows a representation for the digiit "3".

Figure 25: A representation of the letter "S"

Figure 26: A representation of the digit "3".

Note that all of the structural
elements of the two representations are identical. What then leads the
representation for
S to
self-organize into
an "S" while
the representation for 3 self-organizes into a "3"? The difference
is in the length of the invisible-line which connects the mid points
of the two
line segments. The length for "3" is shorter, pulling the two mid
points closer and therefore
on the same side while the length of the invisible-line connecting the
two mid points is longer for "S", pushing them onto opposite sides.
Here is an
web file that allows you to vary the length of that invisible-line. When
you view that web file, that
you can see that shortening the apppriate invisible-line of an "S" makes
it eventually "flip" over to a "3". Lengthening that invisible-line stretches
the entire shape out into
a straight line, which, when you then shorten the invisible-line, reorganizes
into an "S.

With the
components of a simple line, a more complex line, and an invisible line, and
with the
proximity
mediator "near",
we can represent the rest of the letters in the Roman alphabet and other simple
shapes in the same self-repairing,
self-organizing way.

More abstract mediators.

This framework of interactive mediators can be used in more abstract domains
than purely spatial objects. Let us look at some simple examples in the abstract
domain of numbers.

Numeric mediators. Mathematical functions such
as "plus" or "times" are often thought of as little factories,
with two numbers as inputs
and one
number
as an output. So if you give 2 and 3 as inputs to "plus", you get
out a "5".
Let us instead view a mathematical function as an interactive mediator, operating
on all of its immediate neighbors to maintain a specified relationship, where
a change in any of the neighbors leads to changes in all the other neighbors.
So, for the "plus" mediator, if the total increases, then the plus
mediator will operate to increase each of the two addends. This
is shown in Figure 25.

Figure 25: The operation of the "plus" numeric mediator.

Multiple compatible numeric mediators. Just as with the spatial
mediators discussed previously, we can have multiple mediators operating on
a set of concepts, each trying to modify its immediate neighbors to satisfy
its defining relationship. Figure
26 shows two numeric mediators, "plus" and "equal".

Figure 26: The operation of two compatible numeric mediators, plus and equal.

Because these two mediators are compatible, a stable state can be reached
that satisfies both. In the case shown in figure 26, since the activity level
of the "plus" mediator is higher than that of the "equal" mediator,
the satisfaction of the "plus" mediator is reached sooner in time
than that of the "equal" mediator,
as shown in the graph to the lower left. In fact, the equal state was not quite
reached when the screenshot for Figure 26 was taken, so N1 and N2 are still
not quite equal.

Figure 27: The operation of two non-compatible numeric mediators, plus and
times.

If the "plus" mediator has a higher activity level than the "times" mediator,
then the relationship among the three numbers will be closer to an additive
one; if the "times" mediator is more active, then the relationship
will be closer to a multiplicative one. Except for the case in which all three
numbers
are zero, neither of these two relationships will be totally satisfied as long
as the other mediator has a positive activity level. However, the representation
will settle into a state where the more active mediator has more impact and
the less active mediator has a smaller impact.

Key concepts.

There are several concepts that recur throughout the different
domains that have been described in this paper.

Dynamic stability. One is the concept of dynamic
stability. Under the impact of multiple simultaneous mediators, a stable
state can occur that is the result of the dynamic operation of those multiple
mediators. If any of the mediators change, then the stable state changes.
One example of a dynamic stability is the property of the length of a line
segment, which is the dynamic stability reach by the opposition of the two
mediators of repulsion and attraction of landmark subparts of the line segment.

Self-repair. The representations illustrated
here are characterized by the property of "self-repair". That is, they have
a resistance to change, as the multiple simultaneous mediators operate to maintain
the overall structure of the representation. Local changes to components lead
to efforts to restore the overall structure, as each mediator operates on its
immediate neighbors to maintain its specified relationship. While the overall
structure is maintained, the disruptions to the structure often lead to changes
in other global properties. For example, the movement of a subpart of the spatial
representations of letters illustrated above lead to self-repair of the letter's
overall structure, but lead to translation and/or rotation of the letter. In
some cases, a change in a component can lead to a reorganization of the representation,
depending on the nature of the change. Thus the entities being represented
have an overall stability given changes in their environment, but can be modified
if certain kinds of changes occur. This gives them a dynamic stability in overall
structure.

Self-organization. Another property
of this landmark representation and interactive mediation framework is that
of self-organization. Given a random starting point for all the components,
the simultaneous operation of multiple mediators, each operating only on its
immediate neighbors, can lead to the creation of the overall structure of the
representation. This property addresses the issue of how new instances of any
representation are generated in the first place.

Multiple stable states and reorganization. Many
of the representations have more than one stable state, and so some changes
to a representation in one stable state will lead not to self-repair
to that stable state but instead to reorganization into another stable state.
Thus this framework captures the stability that is characteristic of many things,
but also allows for change.

Future directions.

There are a number of different domains in which this framework
shows promise in allowing us to characterize the highly interactive nature
of the interaction that characterizes those domains. These areas of future
development illustrate that the concepts presented here may have applicability
to domains with much larger scales of analysis that the ones presented here.

Interactive mediation and discourse structure. One
domain in which the landmark representation and interactive mediational framework
seems to have promise is in the area of discourse analysis. Topic threads form
the landmarks of discourse, with the continuous"flow" of discourse
characterized by the creation of new topic threads which maintain their activity
level for
a time before fading while new landmark topic threads are created. The content
of the topic threads mediates the creation of meaning among the participants
in the discourse. The mediation provided by the variety of participant roles
moves the discourse, sometimes in concert with each other,
sometimes
in conflict with each other.

Interactive mediation and classroom learning. Another
domain for which this framework has promise is in analyzing the role that the
variety
of artifacts and mediators that are present in any classroom serve in the learning
that occurs within the classroom. The variety of artifacts present in any classroom
serve as landmarks for the interaction, and the variety of mediators self-organize
the
interactional
structure that emerges to support learning among the participants in a classroom.
The mediators in a classroom can be coordinated, operating to accomplish common
goals; they can be uncoordinated, operating independently; or they can be discoordinated,
operating in opposition to each other.

Interactive mediation and organizational reform. Educational
institutions, like other large organizations, exhibit the dynamic stability
property, reacting to reform efforts with the self-repair efforts characterized
by the
operation
of multiple
mediators,
each operating on its immediate neighbors to maintain a specified relationship.
Thus most reform efforts lead to small changes in the overall organization,
which are "self-repaired" leading to a restoration of the overall
structure of the organization with only small resulting global changes. However
some
reform efforts lead to a reorganization of the overall structure of the organization.
The use of this framework may help to determine which reform efforts will lead
to self-repair and which to reorganization, through the characterization of
the landmarks and mediators that characterize the organization. The concept
of mutual attraction can be applied as affiliation among different parts of
an organization; the concept of mutual repulsion can be applied at this level
as disaffection. Status and power operate as mediators to distribute impact
among the components of the organization.

Summary.

These future directions point
to the possibilities that the landmark representation and interactive mediation
framework
presented here
may be applicable to a variety of domains, with a range of scales of analysis.
By using more dynamic media to express
our theories and models, we may be able to better capture the dynamic properties
of systems that have proven difficult to represent in more static media like
print on paper.

Concepts: Their activity, their actions, their interactions,
and their mediation of interaction.

Let us start with the idea of a concept as the basic unit of
representation. A concept is represented as a "node", interconnected
with other concepts. Another key property of a concept is its activity level.
A concept can have a zero activity level (that is, it can be totally inactive),
it can have a high activity level, or it can have any level of activity in
between. Figure A1 shows a concept with zero activity level and the same concept
with a high activity level.

Figure A1: A concept with zero activity level (left) and a
concept with a high activity level (right).

Figure A2: A NetLogo web file that shows the activity level of
a single concept.

What does the activity level of a concept mean? The general idea
is that the higher the activity level a concept has, the more impact it has
on neighboring concepts. What does "impact" mean? Let us look at
some simple kinds of impact that one concept can have on another.

Action of one concept on a second concept.

If we have two concepts, there are three ways that the first
concept can impact the activity level of a second concept: 1) the first concept
can increase the activity level of the second concept, 2) the first concept
can decrease the activity level of the second concept, or 3) the first concept
can have no impact on the activity level of the second concept. These three
possibilities are illustrated in Figure A3. The action of the first concept
to increase the activity level of the second is illustrated by an "excitatory" arrow
from the first concept to the second, shown on the left of Figure A3. The action
of the first concept to decrease the activity level of the second is shown
by the "inhibitory" link (a line with a end-cap), in the center of
Figure A3. The (non)action of the first concept having no impact on the activity
level of the second is shown on the right of Figure A3, with the two concepts
shown with no interconnection.

Figure A3: Three possible actions of one concept on the activity
level of a second concept: excite, inhibit, none.

Figure A4: An web file that shows the impact of one concept on
the activity level of a second concept.

If there is an excitatory link between one concept and another,
then the activity level of the second concept increases when the activity level
of the first concept increases. If there is an inhibitory link between one
concept and another, then the activity level of the second concept decreases
when the activity level of the first concept increases. If there is no link,
then the activity level of the second concept is not directly changed when
the activity level of the first concept changes.

Interaction of two concepts.

With two concepts, there can be not only just the action of one
concept on a second, but also the simultaneous action of the second on the
first. Since there are three possible ways for the second concept to act on
the first for each of the three ways we've just seen for the first to act on
the second, there are nine possible interaction cases, as shown in Table A1.

Figure A5: An web file that shows the interaction of two concepts
with each other

.

Of these nine interaction types, we have already seen five of
them previously: A -> B; A -| B; A B, and the two reverse cases: A <-
B and A |- B which are equivalent to the forward cases. This leaves three new
mutual interaction patterns, shown in Figure A6: 1) mutual excitation A <-> B;
2) mutual inhibition A |-| B; and 3) excitation-inhibition A |-> B and the
equivalent reverse case of A <-| B.

Mutual excitation: At first glance, it looks
like mutual excitation would cause an explosive positive feedback condition
in which the activity levels of both concepts would go to their maximum values.
But in fact this interaction pattern stabilizes fairly quickly, with the activity
levels of both concepts increased. An interesting effect of this interaction
is that initial differences in activity levels between the two concepts are
systematically reduced. In figure A5, you can see that there is a difference
of 4 units of activity level between A and B. With no interaction, the difference
is 7, as shown in figure A7.

Figure A7: Activity levels of concepts A & B with no interaction.

So, overall the impact of mutual excitation is to increase the
activity levels of the participating concepts, but to smooth out differences
in activity levels.

Mutual inhibition: You might imagine at first
glance that the impact of mutual inhibition would be to drive the activity
levels of both concepts to zero. However, again, like mutual excitation, this
interaction pattern stabilizes fairly quickly, with the activity levels of
both concepts reduced but in many cases to a non-zero level. While mutual excitation
reduces differences, mutual inhibition increases differences. With the same
input conditions as shown in figure A7 which produce a difference of 7 in activity
levels, mutual inhibition (as shown in the middle of Figure A6, produces a
difference in activity levels of 20 units. So the overall impact is to decrease
the activity
levels of the participating concepts, but to amplify differences in activity
levels.

Excitation-Inhibition: The third pattern is
the asymmetrical interaction where one concept excites the other, but the other
inhibits the first. As a pair-wise interaction, this also stabilizes fairly
quickly. As far as properties of this interaction pattern, we will revisit
this pattern below in the discussion of mediation among three concepts.

Figure A8: An web file that allows exploration of interaction
among three concepts.

In the example shown in Figure A8, if we have a set three concepts
interconnected with the "excite-inhibit" interaction pattern, we
get an oscillating dynamic equilibrium, in which each concept's activity level
increases and decreases out of phase with the increases and decreases of the
other two concepts. So while two concepts interconnected with this interaction
pattern reaches a stable state, the mediation of a third concept, even when
it has no external activation, creates a dynamic oscillation. This is an example
of the power of mediation in this simplified world.

In this case, we're considering concept C as mediating between
concepts A and B. But the situation is symmetrical - we could just as well
consider concept A as mediating between concept B and concept C, or concept
B as mediating between concept A and concept C. In fact, each is mediating
between the other two at the same time as the others are also mediating. This
idea of multiple simultaneous mediators will recur elsewhere in this paper.

Mediation among four concepts

Mutual inhibition amplifies differences between concepts, as
shown in Figure A6. If we add "context" to those two concepts, then
the ampification takes on a "step-function" character, where the
domination of one concept is sudden and somewhat persistant. Figure A9 shows
the state right after the transition has occurred.

Figure A9: An web file that allows exploration of interaction
among four concepts.

The "saliences" graph shows the gradual decrease in
the activity levels of concepts B and D as the external activity input to B
is decreased from 24 to 17, and the sudden decrease in the activity levels
to zero when the input is decreased from 17 to 16, as the four concepts "flip"
from one state to the other. The same graph shows that increasing the external
activity from 16 back up to 24 doesn't change the activity levels. However,
an increase from 24 to 25 will "flip" the system back to the original
state. In this sense, this system of four concepts exhibits a "memory" effect,
such that a change in the external conditions cause a change that remains when
the external conditions return to their original state.